Belt damping

ABSTRACT

A method of determining damping coefficients for a resilient web, involving: stretching a prescribed length of the web between a pair of pulleys mounted on a frame; attaching a prescribed test mass M o  upon this length at a prescribed distance L 1  from one pulley; shaking the frame at resonance frequency f R , while deriving the resonance-amplitude; and using the foregoing to determine web damping coefficients.

This disclosure relates to belts for transmitting motion (e.g., betweenshafts, with tension transfer and the like), and particularly totechniques for determining damping coefficients therefor.

BACKGROUND, FEATURES

Workers making or using flexible web means (e.g., belts used to transmitmotion from one shaft to another) have long been concerned about thecomplications now typically associated with determining dampingcharacteristics (e.g., damping coefficient) of such a web. Typically,this may involve "trial-and-error" testing of the actual belt in theactual mechanism; or it may require a specific sample size, or it mayrequire that the belt be permanently altered in some way; or"over-stressed" in actual use-environment.

This invention addresses such concerns, and teaches a technique andapparatus for determining such damping coefficients:

without need of trial-error testing of an actual belt length in anactual use-environment, or any associated belt altering oroverstressing;

without need of any specific sample size or belt-length (e.g., testingone length can determine damping for many different lengths).

Thus, it is an object hereof to alleviate such problems and provide atleast some of the here-described features and advantages. A moreparticular object is to provide means for quantifying belt dampingparameters--especially for various belt-lengths, yet by testing at onlya few belt-length positions. Another object is to do so by subjectingthe belt to sinusoidal shaking at resonance conditions.

A further object is to avoid conventional solutions, such as testing abelt in the mechanism it is to be used in.

Other objects and advantages of the present invention will be apparentto those skilled in the art.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other features and advantages of the present inventionwill be appreciated by workers as they become better understood byreference to the following detailed description of the present preferredembodiments, these being considered in conjunction with the accompanyingdrawings, wherein like reference symbols denote like elements:

FIG. 1 is a very schematic, idealized showing of a preferred belttensioning/shaking arrangement;

FIG. 2 illustrates a preferred measurement array for measuring beltdamping for an arrangement like that of FIG. 1; and

FIGS. 3 and 4 illustrate two models for related analysis; and FIGS. 3A,3B give related analysis.

DETAILED DESCRIPTION

FIG. 1 shows a frame F which is relatively rigid when compared to thebelt B being tested, being secured to a shaker table (not shown) adaptedto experience a known sinusoidal motion. Pulleys P-1, P-2 are affixed onthe frame F. A length of belt B (endless-loop or not; also includes anybelt-like, web material) is stretched over, and between, the pulleysP-1, P-2. The distance L between pulleys is adjustable so that a desiredtension may be produced in belt length B. Such tensioning is well knownto those practiced in the related arts. Belt B is clamped to the pulleysover a narrow arc, away from the points where the belt segments aretangent to the pulleys. This prevents gross relative motion between thebelt and the pulleys.

A weight 10 of known mass (test mass m), is clamped to belt B at someknown distance L1 from the center of one pulley P-2. Various known meansof clamping may be used, such as a common C-clamp as known in the art.This clamp, with any needed transducer attached to it, should have itscenter of gravity near the (width, and thickness) axis of belt B.

Motion-Detect means MD is used for measuring the verticalmotion-amplitude of mass m, either relative to the shaker table orrelative to the same reference as the shaker motion. This may beaccomplished by any one of many well known expedients such as: anaccelerometer or like transducer (with indicator also used, ifdesired--see Gig. 2), or a recording pen, or a light beam reflected offthe mass m onto a scale, or an electromagnetic voltage generator, etc.For illustration purposes, accelerometers will be preferred here tomeasure motions of both the mass m and the shaker table. Suchaccelerometers are well known to artisans familiar with motiontransducers. For illustration purposes, these accelerometers are assumedto measure acceleration relative to the earth. Piezo-electricaccelerometers are well known examples and preferred. Or, one mayequivalently measure velocity or displacement, instead of acceleration,as artisans understand.

The belt and frame assembly is mounted to the shaker table so that thelength of the belt L, under test, lies along the same direction as themotion of the shaker table (see arrow FIG. 1). Motion of the shakertable is transmitted to the mass m (along plane of frame F) via pulleysP-1, P-2 and belt B wound thereon.

The motion of mass m may, or may not, have the same amplitude as that ofthe shaker table because of the longitudinal (i.e.,; tension andcompression) flexibility of the belt B. Belt B and object 10 clampedthereon may be viewed as, essentially, the equivalent of a spring andmass system, having a resonant frequency f_(R). It will be assumed thatwhen a spring and mass system is shaken at its resonant frequency, themotion of the mass is limited only by the damping in the system. Here,belt B is assumed to provide the only significant loss of energy in thissystem, thus damping must be, essentially, entirely due to the (dynamic)belt stretching. Here, it will be assumed that a damping coefficient, c,in units of force per unit velocity, can be derived by the followingequation: ##EQU1##

Where:

pi=3.14159

m: mass of the object clamped to the belt B

f_(R) : frequency of sinusoidal motion at resonance

xs: absolute value of motion amplitude for shaker table

xm: absolute value of amplitude for motion of mass m,

System for measuring damping (FIG. 2):

FIG. 2 is a block diagram of a preferred, related instrumentation systemIS for determining belt damping-coefficient C. A tunable sine wavegenerator 2-1 is used to output a sine wave voltage at a selectedfrequency, (this may be monitored by associated frequency meter 2-3, andthis sine wave voltage from generator 2-1 may be input to an amplifier2-5). Generator 2-1 is coupled to drive an electrodynamic shaker tablest. (This may alternatively be a electrohydraulically-driven shakertable.) The shaker table motion is preferably detected by anaccelerometer 2-a (or like amplitude transducer means) whose electricaloutput may be fed (e.g., via an amplifier 2-7 if needed), to anamplitude meter 2-9 which may indicate the value xs (shake-amplitude fortable st) if desired.

Similarly, an accelerometer 20-aa is attached to the mass m (that isclamped onto the belt, and its output may be fed to an amplifier 20-1,and then to an amplitude meter 20-3 if desired, to indicate the value xm(shake-amplitude for mass m).

In any event, each accelerometer output is fed, in common, to a phasemeter 20-5, which measures the relative phase between the peaks of the(sine wave) motions of the mass and table while shaken. The tuneablesinewave generator 2-1 is adjusted until these relative phases are at90° degrees.

This 90° degree phase relation is here assumed to occur at "resonance".One example for so measuring phase is with the use of Lisajou figures onan oscilloscope. Another way to measure phase, as well as amplitude, iswith an FFT analyzer. These devices are in common use. The frequency atwhich the 90° degree phase relation occurs is, of course, resonantfrequency: the value f_(R) used in the above equation.

Note: the equation above for determining value C has no dependence onbelt length. If mass m is clamped at a different position along the beltlength, and the measurement process repeated, a different value of cwill result. It will be shown that this should occur by reference(below), to FIG. 3, and will use this as a straight-forward"stress-strain, tension/compression bar model" to deriveforce-displacement equations in terms of elastic and damping parameters.When compared with the force-displacement equation for the "lumpedparameter" model, FIG. 4, it will be seen that any segment of such abelt should have a damping coefficient which is inversely proportionalto belt length (i.e.,c˜1/L).

Analysis per Tension-Compression BAR Model (FIG. 3):

FIG. 3 may be understood as a "BAR model" for analyzing forces F,F'stressing a subject length of such a belt B in tension or compression.The segment will be assumed to have a length L with a test mass mdisposed therealong at varied positions x, and the resultantdisplacements (under sinusoidal shaking at resonance) will be U_(o),U_(e) at respective ends.

Assuming that strain ε is defined as du/dx, a displacement U may bedefined as the integral; ##EQU2## Stress σ is given as: σ=E ε+qdE/dt,orσ=F/A Where

ε=elastic modulus,

q=damping coefficient for bar

t=time

A=cross-sectional area

Thus, one may represent differential motion, U_(L) -U₀ as: ##EQU3##above analysis summarized in FIG. 3a.

Workers will here understand that a belt is really a tension-compressionbar model. The Test Fixture, here mentioned, yields data for a "LumpedParameter" model (e.g., see below, and FIG. 4). The following analysisof the Lumped Parameter model shows how the measured damping data can berelated to the "BAR" model.

Analysis Via "Lumped-Parameter" model (FIG. 4):

FIG. 4 may be understood as a "Lumped-Parameter" model, to be comparedwith the BAR model above, Here, cc represents a "dashpot dampingcoefficient, and K: spring stiffness, with U_(o),U_(e) as before.

At dashpot dd, the damping force F_(cc) may be expressed as: ##EQU4##

At "spring" k, related force F_(k) may be expressed as:

    F.sub.k =K (u.sub.e -u.sub.o)

Total force F is F_(cc) +F_(k)

Hence, the differential displacement u_(e) -u_(o) is:

    u.sub.e -u.sub.o =F/k (i-e.sup.KT/c)

Comparing with the BAR model:

k=AE_(/L)

c=Aq:_(/L)

Thus, damping coefficient C is inversely proportional to L, the lengthof belt being stretched (c˜1/L).

Since the belt in FIG. 1 has two segments, L1 and L-L1, which areundergoing tension/compression, the related damping coefficient cmeasured is actually the sum of the two different damping coefficients:C₂ /L₁ and C₂ /L-L₁, where C₂ =A_(q). Additionally, there may be dampingat the interfaces where the belt is tangent to the pulleys. This isparticularly true for toothed synchronous belts where a belt tooth mayrub on a pulley groove during this test. This damping is a constant,since it is not dependent on the position of the clamped mass. Themeasured damping coefficient c can be described as follows:

    c=2*c1+c2/L1+c2/(L-L1);

or,

    c=2c.sub.1 +c2/L.sub.1 +c2/L-L.sub.1

To obtain the damping coefficients c1 and c2, tests are run for severalpositions (several L1 values). If only two positions are used, then onecan employ a process for solving two linear equations and two unknowns,and so calculate the constants c1 and c2. If more than two positions areused, then one of several well known "least squares, curve fitting"processes can be used to find the "best" values for c1 and c2. Anexample of such a process is the Marquardt-Levenbert algorithm. ofcourse, using more test positions gives more accurate values for c₁ andc₂.

This invention will be thus understood to measure damping parameters forresilient belts such that a lumped parameter damping coefficient can beeasily calculated for any length of belt (used to transmit motion fromone shaft to another).

This is done, principally to determine belt damping coefficients, and soenable an accurate prediction of a belt's dynamic motionperformance,--for mechanisms utilizing such belts, so that differentbelts can be compared for their inherent damping capability, and so thata belt can be "sized" to obtain desirable damping properties.

A salient advantage is that belt damping can be determined withoutactually testing the belt in the mechanism for using it. Also, one canpredict damping for belts of different lengths just by testing onelength of belt. A simple test fixture can be used for many differentkinds and lengths of belt construction. One can also so test otherresilient "belt-like" web materials, such as lengths of: photographicfilm, magnetic tape, ink ribbon, rope, cables, paper strips, fabricstrips, etc.

Advantages Over Past Practice:

--Does not require "trial-and-error" testing in the actual mechanism;and just one belt-length is needed to determine damping for manydifferent lengths;

--Does not require a specific sample size, as virtually any length beltmay be used.

--Does not require that the belt be permanently altered in any way; or"over-stressed" or tested in actual use-environment.

Related products for using such belts or belt-like materials are:microfilm film advance mechanisms, document transports, documentpositioning systems, paper advance mechanisms in printers, pen plotters,magnetic and optical digital storage devices, magnetic tape recorders,printhead positioning mechanisms in typewriters and computer printers,printer ribbon advance mechanisms, optical mirror positioningmechanisms, robots, automatic assembly mechanisms, automotive alternatordrives, camshaft drives, and air conditioning compressor drives,automatic adhesive tape dispensers. Such can also benefit from thisinvention.

In summary, the damping coefficient of a resilient web length isdetermined by:

1--stretching the length between a pair of pulley means, separated bydistance L, and mounted on a relatively fixed frame; with a prescribedtest mass m clamped thereon at distance L₁ from a pulley;

2--mounting the frame on a shaker table and providing a shaking means(pref. sinusoidal) to shake the frame and test mass whilemonitoring/adjusting the frequency thereof to arrive at resonance;

3--deriving accelerometer output (amplitude) from this table as it isshaken, while measuring this amplitude A_(T) at resonance;

4--deriving accelerometer output (amplitude A_(m)) from this test massm, while measuring this amplitude A_(m) at resonance;

5--adjusting the generator to shake the table and the mass at resonantfrequency f_(R) ;

[e.g., doing so via phase monitor, imposing orthogonal phase-relation]

6--computing damping constant C₁ and C₂ from measurements at two or moremass positions (e.g., L₁);

7--and so determining a damping coefficient C according to the relation;

    C=2C.sub.1 +C.sub.2 /L.sub.1 +C.sub.2 /L-L.sub.1, etc.

Of course, many modifications to the preferred embodiment described arepossible without departing from the spirit of the present invention. Theinvention is not limited to the particular types of sensors or shakersor mountings. Additionally, some features of the present invention canbe used to advantage without the corresponding use of other features.

Accordingly, the description of the preferred embodiment should be to beconsidered as including all possible modifications and variations comingwithin the scope of the invention as defined by the appended claims.

What is claimed is:
 1. A method of determining damping coefficients fora resilient web, comprising:stretching a prescribed length of said webbetween a pair of pulley means mounted on frame means; attaching aprescribed test mass M_(o) on said length at a prescribed distance L₁from one said pulley means; shaking said frame means at resonancefrequency f_(R), while deriving resonance-amplitude; and using theforegoing to determine damping coefficients.
 2. The method of claim 1,wherein said distance L₁ is varied N times, with a damping coefficientdetermined for each distance L₁, and these coefficients reconciled. 3.The method of claim 2, wherein said frame is affixed on shake-surfacemeans, and controllable shaking means is applied to so shake said framemeans and surface means at resonance frequency.
 4. The method of claim3, wherein said shake means is controlled to execute sinusoidal shakevibration.
 5. The method of claim 4, wherein the resonant amplitudes ofsaid surface means and said test mass M_(o) are derived.
 6. The methodof claim 5, wherein accelerometer and transducers are used to indicatesaid amplitudes.
 7. The method of claim 6, wherein phase is monitored todetermine resonance.
 8. The method of claim 7, wherein N differentmass-distances L₁ are selected and damping coefficient determined foreach, with an overall damping coefficient for the belt derivedtherefrom.
 9. The method of claim 1, wherein said web comprises a beltfor transmitting motion.
 10. The method of claim 9, wherein shaking isperformed by tuneable sine-wave generator means.
 11. The method of claim8, wherein a damping coefficient C is derived according to the relation:##EQU5## Where: pi=3.14159m: mass of the object clamped to the belt Bf_(R) : frequency of sinusoidal motion at resonance xs: absolute valueof motion amplitude for shaker table xm: absolute value of amplitude formotion of mass m.
 12. An arrangement for determining dampingcoefficients for a resilient web, comprising:frame means with pulleymeans for stretching a prescribed length of said web there-between; saidpulley means mounted on said frame means; a prescribed test mass M_(o)mounted on said length at a prescribed distance L₁ from one said pulleymeans; shake means for shaking said frame means at resonance frequencyf_(R), sensor means for deriving the resonance-amplitude of said massand said frame means, with the foregoing being used to determine dampingcoefficients for each distance L₁.
 13. The arrangement of claim 12,where said distances L₁ are varied and a damping coefficient determinedat each distance.
 14. The arrangement of claim 13, where said frame isaffixed on shake-surface means, and controllable shaking means isapplied to so shake said frame means and surface means, plus said mass.15. The arrangement of claim 14, where said shaking means is arranged toexecute sinusoidal vibration.
 16. The arrangement of claim 15, whereshake-transducers are coupled to said mass and said frame/surface meansto indicate the resonance said amplitudes.
 17. The arrangement of claim16, where phase detect means is used to determine resonance.
 18. Thearrangement of claim 17, where said web comprises a belt fortransmitting motion.
 19. The arrangement of claim 18, where tuneablesine-wave generator means is used for said shaking.
 20. The arrangementof claim 19, where a damping coefficient C is derived according to therelation: ##EQU6## Where: pi=3.14159m: mass of the object clamped to thebelt B f_(R) : frequency of sinusoidal motion at resonance xs: absolutevalue of motion amplitude for shaker table xm: absolute value ofamplitude for motion of mass m.
 21. The arrangement of claim 15, whereintransducer means are affixed to said test mass and said framemeans/surface means, with the outputs thereof are fed in common, tophase detect means whereby to indicate shake-amplitude at resonance.